It's just so stupid. If anyone in his inner circle gets caught, they can immediately be discovered to be a Death Eater and sentenced, no other evidence needed. Is there any reasonable explanation for why he did this?

I understand that he used it to send messages, but why didn't he use something else that wasn't a tattoo of his own sign, telling everyone that saw it that the person was a Death Eater?

Not really an answer as such, but worth remembering that the Dark Marks were invented and branded on to the Death Eaters at a time when Voldemort was not quite so sly and working in secret as during the events of OotP. Also, considering the fact that statements and testimonies seemed to play the central part in determining whether Death Eaters were convicted or not, it seems likely that either a) there were non-Death Eaters around who’d had the Mark branded onto them (either under the Imperius curse or through coercion), or b) the Mark could be somehow hidden at will when the bearer wished.
– Janus Bahs JacquetFeb 5 '16 at 23:58

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@Janus Bahs Jacquet- I wouldn't base anything on the trials, considering that the Mark disappeared when Voldemort did. It only reappeared when he regained strength.
– TenthJusticeFeb 6 '16 at 1:12

@Tenth But most likely there were trials before Voldemort was caught as well.
– Janus Bahs JacquetFeb 6 '16 at 1:19

I always imagined that it relates to the branding of cattle. It's his way of saying: "I own you, you are cattle to me." Moo.
– pleurocoelusFeb 6 '16 at 16:32

It's not worth a separate answer, but no one has mentioned yet Voldemort's arrogance. He wouldn't want to have anything similar to the Dark Mark but have it a random design for discretion. Often it is made clear that simple, megalomaniacal arrogance led to his undoing. This is just another example.
– ThruGogFeb 7 '16 at 4:43

3 Answers
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Nobody at the Ministry knew. They knew what the Dark Mark meant, of course, but they didn't know Voldemort was branding his inner circle with it. Nobody knew so nobody looked.

Snape strode forwards, past Dumbledore, pulling up the left sleeve of his robes as he went. He stuck out his forearm, and showed it to Fudge, who recoiled.

...

Fudge stepped back from Snape, too. He was shaking his head. He did not seem to have taken in a word Snape had said. He stared, apparently repelled, at the ugly mark on Snape's arm, then looked up at Dumbledore and whispered, 'I don't know what you and your staff are playing at Dumbledore, but I have heard enough. I have no more to add.'

Harry Potter and the Goblet of Fire - p.616 - Bloomsbury - Chapter 36, The Parting of the Ways

Now, I know Fudge is in full denialist mode right now, but the Minister of Magic himself does not seem to have any notion of what that brand truly indicated. There is no evidence that the matter of the brand ever came up during Death Eater trials and I conclude that nobody at the ministry was aware of Voldemort's practice of branding his Death Eaters.

Now of course if anybody had rolled up a prisoner's sleeve that might have created issues - why are you wearing Voldemort's mark as some kind of demented tattoo? - but the Death Eaters were smart enough to largely avoid capture prior to Voldemort's downfall - at which point the mark had faded anyway.

As for whether it was such a good idea in the event that a Death Eater got caught in service - well, I don't think Voldemort really wanted his Death Eaters denouncing him to stay out of prison, so I'm not really sure what you think he stands to lose by branding them. In fact, if anything, he seems to be trying to enforce loyalty. Just witness the scene in the graveyard:

'... we are still united under the Dark Mark, then! Or are we?'

...

'I smell guilt,' he said. 'There is a stench of guilt upon the air.'

...

'I see you all, whole and healthy, with your powers intact - such prompt appearances! - and I ask myself ... why did this band of wizards never come to the aid of their master, to whom they swore eternal loyalty?'

...

'And I answer myself,' whispered Voldemort, 'they must have believed me broken, they thought I was gone. They slipped back among my enemies, and they pleaded innocence, and ignorance, and bewitchment ...'

'It is a disappointment to me ... I confess myself disappointed ...'

'Stand up. You ask for forgiveness? I do not forgive. I do not forget. Thirteen long years ... I want thirteen years' repayment before I forgive you.'

...

'And yet you ran from my Mark, when a faithful Death Eater sent it into the sky last summer? ... Yes, I know all about that, Lucius ... you have disappointed me ... I expect more faithful service in future.'

...

'The Lestranges should stand here,' said Voldemort quietly. 'But they are entombed in Azkaban. They were faithful. They went to Azkaban rather than renounce me ... when Azkaban is broken open, the Lestranges will be honoured beyond their dreams.'

The first thing to remember is that most of the Death Eaters who escaped punishment after the First Wizarding War simply claimed to be under the Imperious Curse. So the Dark Mark isn't enough evidence on its own of wrongdoing, you'd have to prove they were a willing participant. (remember, even the Imperioused Minister of Magic Pius Thicknesse is made a Death Eater)

And assuming Wizarding Laws are similar to our own... what does having a Dark Mark actually prove? In most legal systems, a gang tattoo and even professed gang affiliation is not sufficient to send someone to prison. You need evidence of specific involvement in a specific crime.

But the peril of easy identification was outweighed by the usefulness of the Dark Mark, which had several magical properties beyond a simple tattoo:

The Dark Mark:

Allowed Death Eaters to "distinguish one another," per Snape. This is actually pretty necessary, since Karkaroff says during his trial that only Voldemort knew the identity of every other Death Eater. By burning it into one's skin instead of something like a coin, this also avoided the situation where an Auror poses as a Death Eater.

Summoned Death Eaters to his exact location. Since Voldemort couldn't stay in one place too long, this would be extremely useful.

Allowed them to summon him. Considering Voldemort is the only one who can kill the Chosen One, this is vital.

Allowed for easy intimidation. Draco uses it to cow Mr. Borgin into helping him in Half-Blood Prince.

Allowed Death Eaters to bypass a special barrier spell that (presumably) Voldemort taught them.

"I saw [Snape] run straight through the cursed barrier as though it wasn't there," said Lupin. "I tried to follow him, but was thrown back just like Neville. . .

"He was," said Harry savagely, "but to help them, not to stop them . . . and I'll bet you had to have a Dark Mark to get through that barrier...

Acted as a sort of status symbol, allowing Voldemort to signify his most trusted advisers. Fenrir Greyback is taken much less seriously do to his lack of a Dark Mark, while Draco getting one is a huge deal.

This is just speculation on my part, but due to the repeated emphasis that only Death Eaters can cast the Dark Mark into the sky, I would guess that having the Mark would be a prerequisite.

The key thing to remember here is that whatever steps were taken to keep the Mark secret, they apparently worked. After Voldemort's return, Snape has to explain to the freaking Minister of Magic (a former Department of Law Enforcement employee) about the Dark Mark on his arm and what it means.

"There," said Snape harshly. "There. The Dark Mark. It is not as clear as it was an hour or so ago, when it burned black, but you can still see it. Every Death Eater had the sign burned into him by the Dark Lord. It was a means of distinguishing one another, and his means of summoning us to him. When he touched the Mark of any Death Eater, we were to Disapparate, and Apparate, instantly, at his side. This Mark has been growing clearer all year. Karkaroff's too.

It's more than just a mark, it's a method of summoning, a form of communication. For the first time he rose to power we see his followers proudly wearing it. In a world with the Imperius curse, it's easy to say "Oh that thing? I was bewitched to wear it", which was the case for a lot of things.